9 A linear inequality describes a region of a coordinate plane called a half-plane. All points in the region are solutions of the linear inequality. The boundary line of the region is the graph of the related equation.

11 Graphing Linear InequalitiesStep 1Solve the inequality for y (slope-intercept form).Step 2Graph the boundary line. Use a solid line for ≤ or ≥. Use a dashed line for < or >.Step 3Shade the half-plane above the line for y > or ≥. Shade the half-plane below the line for y < or y ≤. Check your answer.

12 The point (0, 0) is a good test point to use if it does not lie on the boundary line.Helpful Hint

13 Example 2A: Graphing Linear Inequalities in Two VariablesGraph the solutions of the linear inequality.y  2x – 3Step 1 The inequality is already solved for y.Step 2 Graph the boundary line y = 2x – 3. Use a solid line for .Step 3 The inequality is , so shade below the line.

14  Example 2A Continued Graph the solutions of the linear inequality.y  2x – 3Substitute (0, 0) for (x, y) because it is not on the boundary line.Check y  2x – 3(0) – 30 –3A false statement means that the half-plane containing (0, 0) should NOT be shaded. (0, 0) is not one of the solutions, so the graph is shaded correctly.

17  Example 2B Continued Graph the solutions of the linear inequality.5x + 2y > –8Substitute ( 0, 0) for (x, y) because it is not on the boundary line.Checky > x – 4(0) – 40 –4>The point (0, 0) satisfies the inequality, so the graph is correctly shaded.

27 Graph the solutions of the linear inequality.Check It Out! Example 2cGraph the solutions of the linear inequality.Step 1 The inequality is already solved for y.Step 2 Graph the boundary line Use a solid line for ≥.=Step 3 The inequality is ≥, so shade above the line.

28 Check It Out! Example 2c ContinuedGraph the solutions of the linear inequality.Substitute (0, 0) for (x, y) because it is not on the boundary line.Checky ≥ x + 1(0) + 10 ≥ 1A false statement means that the half-plane containing (0, 0) should NOT be shaded. (0, 0) is not one of the solutions, so the graph is shaded correctly.

29 Example 3: ApplicationAda has at most 285 beads to make jewelry. A necklace requires 40 beads, and a bracelet requires 15 beads.a. Write a linear inequality to describe the situation.Let x represent the number of necklaces and y the number of bracelets.Write an inequality. Use ≤ for “at most.”

31 Example 3 Continuedb. Graph the solutions.=Step 1 Since Ada cannot make a negative amount of jewelry, the system is graphed only in Quadrant I. Graph the boundary line Use a solid line for ≤.

32 Example 3 Continuedc. Graph the solutions.Step 2 Shade below the line. Ada can only make whole numbers of jewelry. All points on or below the line with whole number coordinates are the different combinations of bracelets and necklaces that Ada can make.

33 Example 3d. Give two combinations of necklaces and bracelets that Ada could make.Two different combinations of jewelry that Ada could make with 285 beads could be 2 necklaces and 8 bracelets or 5 necklaces and 3 bracelets.(2, 8)(5, 3)

34 Check It Out! Example 3What if…? Dirk is going to bring two types of olives to the Honor Society induction and can spend no more than $6. Green olives cost $2 per pound and black olives cost $2.50 per pound.a. Write a linear inequality to describe the situation.b. Graph the solutions.c. Give two combinations of olives that Dirk could buy.

35 Check It Out! Example 3 ContinuedLet x represent the number of pounds of green olives and let y represent the number of pounds of black olives.Write an inequality. Use ≤ for “no more than.”Greenolivesblackplusis no more thantotalcost.2x+2.50y≤6Solve the inequality for y.2x y ≤ 6–2x –2xSubtract 2x from both sides.2.50y ≤ –2x + 6Divide both sides by 2.50.2.50y ≤ –2x + 62.50

36 Check It Out! Example 3 Continuedy ≤ –0.80x + 2.4Green OlivesBlack Olivesb. Graph the solutions.Step 1 Since Dirk cannot buy negative amounts of olive, the system is graphed only in Quadrant I. Graph the boundary line for y = –0.80x Use a solid line for≤.

37 Check It Out! Example 3 Continuedc. Give two combinations of olives that Dirk could buy.Two different combinations of olives that Dirk could purchase with $6 could be 1 pound of green olives and 1 pound of black olives or 0.5 pound of green olives and 2 pounds of black olives.Black Olives(1, 1)(0.5, 2)Green Olives

38 Example 4A: Writing an Inequality from a GraphWrite an inequality to represent the graph.y-intercept: 1; slope:Write an equation in slope-intercept form.The graph is shaded above a dashed boundary line.Replace = with > to write the inequality

39 Example 4B: Writing an Inequality from a GraphWrite an inequality to represent the graph.y-intercept: –5 slope:Write an equation in slope-intercept form.The graph is shaded below a solid boundary line.Replace = with ≤ to write the inequality

42 Lesson Quiz: Part I1. You can spend at most $12.00 for drinks at a picnic. Iced tea costs $1.50 a gallon, and lemonade costs $2.00 per gallon. Write an inequality to describe the situation. Graph the solutions, describe reasonable solutions, and then give two possible combinations of drinks you could buy.1.50x y ≤ 12.00